TL DR; We’ve made a demo snark for Plasma Cash and Cashflow history compaction, that reduces a proof size from ~700 bytes
per block to ~200 + 8*N
bytes for N
blocks
Details
The SNARK itself is located here. In it’s core it proves that a subtree in a block’s SMT is empty for each root hash in the public inputs. Peddersen hash is used as a tree hash, with leaf values being 256 bit strings.
Inputs:
 Start of the continuous noninclusion range
 Length of the continuous range. Currently there is no internal check that start of the interval is divisible by the length of the range, but it’s trivial to extend
 Set of N root hashes for the block’s SMTs for which noninclusion is proved. Those are not compacted using the hash to reduce the number of inputs because a verification of such SNARK will be done offchain only
Workflow:
 Alice accumulated N noninclusion proofs from operator and prepares a SNARK proof as mentioned above
 Upon the transfer Alice sends a proof along with a list of block number for which this proof is valid to Bob
 Bob queries the chain to obtain root hashes for these blocks and use them as public inputs in verification
 Bob accepts a noninclusion proof for this range of block numbers if SNARK proof is valid
Some numbers

4_270_718
constraints for128
block of noninclusion for24
tree depth  provable in
~30 seconds
on my laptop with six core i9  Proof size is
192 bytes
(Groth16 for BN256) +N*8
bytes to encode block numbers to send to Bob (uint64 for block numbers) +8 bytes
for a start of the slice being proved +8 bytes
for a length of the slice
Tradeoffs
 Will require to use Peddersen hash for a tree, that is slower (~
30 microseconds
per round 2n > n)  Will require to use another zkSNARK to prove inclusion upon exits, that is more expensive, but can be batched. Batch size of 5 gives verification cost of 300k gas per proof
Improvements
 May transpose a zkSNARK to prove not the fact of noninclusion of some range for a set of blocks, but a noninclusion of the set of slices in one or more blocks
 Can be trivially extended to sumtrees